143. It sometimes happens, that the numerator, or the denominator, or both, are not perfect squares. If the numerator only is not a square, the approximate root is extracted by the method already explained; and having extracted the root of the denominator, we give to it for numerator, the root of the numerator. Thus, if the square root of is required, we extract the approximate root of the numerator 2, which is 1,4, or 1,41, or 1,414, or 1,4142, &c., according as the root is required to a greater or less degree of exactness. And as the square root of 9 is 3, we have for the approximate root of, the quantity 1,41 1,4142 But if the 3 denominator is not a square, both terms of the fraction are multiplied by the same denominator; (which does not change the value) the denominator is thus made a square, and the operation performed as in the preceding case. For example, if the square root of 3 is required, we change this fraction to 1; extracting the square root of 15, to three places of decimals, for instance, we have 3,872; and as the square root of 25 is 5, the square root 15 will be 3.872 1.4 or 3 or 1,414 or 5 144. In order to avoid fractions of different kinds in the same sum, we reduce the result 3,872, to decimals, by dividing 3,872 by 5, which gives 0,774 for the root of expressed in pure decimals (99.) 145. If whole numbers are joined to fractions, the whole numbers are reduced to fractions (86), and the operation performed according to the direction given for extracting the roots of fractions. Thus, to extract the square root of 83, we change 83 into 59, and this again into 413 (143); of which the approximate root is 20,322 7 9 or 2,903. 146. The fraction, which accompanies the whole number, may also be reduced to decimals; but it is necessary to observe, that the number of decimals employed must be even, and twice as great as the number of decimal figures required in the root. This preparation is made by placing after the given sum 1, 3, or 5, &c. ciphers, which does not change its value (30). Thus, to extract the square root of 21,935 to three places of decimals, M we extract the square root of 21,935000, which is 4,683; this is also the root of 21,935. In like manner, the square root of 0,542, expressed in three decimal figures, is 0,736; and that of 0,0054, to the same value 0,073. 147. When, by the method already explained, we have found the three first figures of the root, many others may be found, with greater facility, by division only, in the following manner. Take, for example, the number 763703556823; we commence by seeking the three first figures of the root as in the preceding manner, and find 873 for the root, and 1574 for remainder. By the side of this remainder we place 55, the two figures, which follow 763703, the part, which gave the three first figures of the root. (If four figures of the root have been found, three figures are placed by the side of the remainder; if five figures of the root have been found, the four following figures are joined to the remainder; and so on.) We then divide 157455 by 1746, twice the root found, and have 90 for the quotient: these are two new figures of the root, which has now become 87390. We square this root, and subtract its square 7637012100, from the part 7637035568, of which 87390 is the root; and there remains 23468. If other figures of the root are required; as we have already obtained five, the four next may be found by division only. For this purpose, we place after the remainder 23468, the remaining figures, 23, of the given number, and two ciphers, and divide 234682300, by 174780, twice the root already found, and have 1342, for the four new figures of the root. But by separating the given number into divisions, according to the requirement of the rule, it is plain, that the root should contain only six figures of whole numbers; the root is then 873901,342, to three places of decimals. The abbreviation, which has been explained, is a consequence of the general principle, which it is easy to deduce from what has been said in article (134), that the square of any quantity composed of two parts, contains the square of the first part, twice the first part multiplied by the second, and the square of the second. EXAMPLES FOR PRACTICE. 1. What is the square root of 152399025 ? Ans. 12345. 2. What is the square root of ,00032754? Ans. ,01809. 3. What is the square root of? Ans.,645497. 4. What is the square root of 6? Ans. 2,5298, &c. 5. What is the square root of 10? Ans. 3,162277, &c. 6. What is the square root of 234,09 ? Ans. 15,3. 7. What is the square root of 1030892198,4001? Ans. 32107,51. 8. A general has an army of 4096 men; how many must he place in rank and file to form them into a square? Ans. 64. 9. A wall is 36 feet high, and a ditch before it is 27 feet wide; what is the length of a ladder, that will reach to the top of the wall from the opposite side of the ditch? Ans. 45 feet. 10. It is required to lay out 25600 square rods of land in a square; what is the side of a square that shall con tain the land? Ans. 160 rods. 11. A gentleman purchased 3025 tiles, for the purpose of paving a square; how many tiles will there be upon a side? Ans. 55. 12. A certain general commanded an army of 49284 men; and the better to secure his standard, he gave orders to form into a square body, the men standing 4 feet distant; what number of men will be on a side; and what quantity of land will they occupy? Ans. 222 men occupy 17,28 acres. 13. What length of rope must be tied to a horse's neck, that he may graze upon 7854 square feet of new feed every day, for 4 days; the end of the rope being each day fastened to the same stake? To perform this and similar questions, divide the given area by ,7854 and the square root of the quotient will be the diameter of the circle fed over by the horse; half of this diameter will be the length of the rope required. Hence, the first rope will be 50 fect, the second 70,5, the third 86,5 and the fourth 100 feet in length. 14. Four men, A, B, C, and D, purchased a large grindstone, the diameter of which was 200 inches; they agreed that D should wear off his share first, and that each man should have it alternately till they had worn off their shares; how much must each man round the stone? wear off Ans. D. 13; C. 16; B. 201; A. 50 inches. NOTE.-The diameter of a circle being given, the area is found by squaring the diameter, and multiplying its square by ,7854. 15. What is the square root of 106929 ? Ans. 327. 16. What is the square root of 36372961 ? Ans. 6031 17. What is the square root of 3271,4007 ? Ans. 57,19, &c. 18. What is the square root of 10,4976? Ans. 3,24. 19. What is the square root of 7056 20. What is the square root of 2704 21. What is the square root of 3739 22. What is the square root of 27% 4225 36? Ans.. Ans. Ans. 64. Ans. 51. 23. If 484 trees are planted at an equal distance from each other, so as to form a square orchard, how many will be in a row each way? Ans. 22. 24. A certain number of men gave 30s. 1d. for a charitable purpose; each man gave as many pence as there were men: how many men were there? Ans. 19. 25. There is a circle whose diameter is 4 inches, what is the diameter of a circle 4 times as large? Ans. 8 inches. NOTE-Square the given diameter, multiply this square by the given proportion, and the square root of the product will be the diameter required. If the required be less than the given diameter, divide the square of the given diameter by the proportion. 26. There are two circular ponds in a gentleman's pleasure grounds; the diameter of the less is 100 feet, and the greater is three times as large. What is its diameter ? Ans. 173,2 &c. 27. If the diameter of a circle be 12 inches, what will be the diameter of another circle, half as large ? Ans. 8,48, &c. inches. 28. A certain castle, which is 45 yards high, is surrounded by a ditch 60 yards broad; length of a ladder to reach from the to the top of the castle? what must be the outside of the ditch Ans. 75 yards. Wall. Ladder. Ditch. NOTE. This diagram is called a right angled triangle; and the square of the hypothenuse, or longest side, is equal to the sum of the squares of the two other sides. Now if the square root of the sum of the squares of the height of the wall and breadth of the ditch be extracted, ive shall have the length of the ladder. If the difference of the squares of the longest and either of the other sides be taken, we shall have the square of the remaining side; the square root of this square will be the length of the side. 29. A line 27 yards long will exactly reach from the top of a fort to the opposite bank of a river, which is known to be 23 yards broad; what is the height of the fort? Ans. 14,142, &c. yards. 30. Suppose a ladder 40 feet long, be so planted as to reach a window 33 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window on the other side 21 feet high; what is the breadth of the street ? Ans. 56,60, &c. feet. FORMATION OF CUBE NUMBERS, AND THE EXTRACTION OF THEIR ROOTS. 140 What is called the cube of a number, is formed by multiplying the number by itself, and afterwards multiplying the product of this multiplication by that same number. The cube of a number, properly speaking, is the product of the square of a number, multiplied by that number. Thus, the cube of 3 is 27, because it is what results from the multiplication of 9, the square of 3, by 3 |